Growth and decay

Multiply by the same number each time

Growth by 4% a year: × 1.04 each year.

A population of 12,000 growing by 4% a year is 12,000 × 1.04⁵ = 14,599.8..., about 14,600, after 5 years. A fall of 10% a year is × 0.9 each year.

Doubling every 20 minutes: × 2 for each 20 minutes. First count the periods: 2 hours = 120 minutes = 6 lots of 20 minutes, so × 2⁶ = × 64, not × 12. 50 cells would become 50 × 64 = 3200.

A half-life is the time it takes for an amount to halve. A half-life of 6 hours: after 18 hours, three halvings: × (1/2)³ = ÷ 8. So 40 g becomes 40 ÷ 8 = 5 g.

To find when a value first falls below a target, try whole numbers of years (or weeks), as with compound interest: 1000 × 0.9ⁿ is 590.49 for n = 5 and 531.441 for n = 6, so it first falls below 550 after 6 years.

An iteration formula says the same thing step by step: Pₙ₊₁ = 1.04Pₙ means "next value = 1.04 × this value". P₀ is the starting value, P₁ the value after one step, and so on. With P₀ = 500: P₁ = 1.04 × 500 = 520, P₂ = 1.04 × 520 = 540.8, P₃ = 1.04 × 540.8 = 562.432. That is the same as 500 × 1.04³.

The shape of the graph tells you the kind of change. Adding the same amount each year gives a straight line. Growing by the same percentage each year gives a curve that gets steeper and steeper (exponential growth). Falling by the same percentage gives a curve that falls steeply, then levels off (exponential decay).

The percentage change from one value to the next is change ÷ original value × 100. From 1,250 to 1,400 the change is 150, and 150 ÷ 1250 × 100 = 12%.